is well-conditioned and hence the variables \((\mathbf {q}^{k_{m+1}}-\mathbf {q}^{k_m})\)  and \((\mathbf {\eta }^{k_{m+1}})_{{\mathcal {A}}_{k_m}}\)  can be computed easily once \(\mathbf {r}^{k_{m+1}}\)  is obtained.
We are thus inspired to choose the Schur complement reduction in [1]}, which is a major segregated approach, to convert the Newtonian system (REF ) to a linear system with only respect to the variable \(\mathbf {r}^{k_{m+1}}\) , by using the block factorization of the coefficient matrix \(F^{k_m}\)  in (REF ). For the matrix \(F^{k_m}\) , it can be factorized as
\(F^{k_m}=\begin{pmatrix}I &\frac{1}{\beta }N_{k_m}^\top M^{-1}&R_{k_m}\\0&I&0\\0&\frac{1}{\beta }{\mathcal {P}}_{{\mathcal {A}}_{k_m}}M^{-1}&I\end{pmatrix}\begin{pmatrix}S_{k_m}&0&0\\0 &\beta M&0\\0&0&-\frac{1}{\beta }({\mathcal {P}}_{{\mathcal {A}}_{k_m}}M^{-1}{\mathcal {P}}_{{\mathcal {A}}_{k_m}}^\top )\end{pmatrix}\begin{pmatrix}I &0&\\-\frac{1}{\beta }M^{-1}N_{k_m}&I&\frac{1}{\beta }M^{-1}{\mathcal {\mathcal {P}}_{{\mathcal {A}}_{k_m}}}\\-R_{k_m}^\top &0&I\end{pmatrix},\) 
